Asymptotic mean density of sub-unitary ensembles

The asymptotic behaviour of the mean spectral density for the product of N × N random unitary matrices distributed according to the Haar measure and a fixed diagonal matrix is investigated. The large N limit of the exact expression derived by Wei and Fyodorov ( 2008 J. Phys. A: Math. Theor. 41 502001 ) is calculated by a modification of the saddle point method. It is shown that in the bulk the result coincides with the one obtained within the free probability theory by Haagerup and Larsen (2000 J. Funct. Anal. 176 331). Close to the edge points of the bulk asymptotics, the mean density for large N is described by a universal function depending on a certain scaling variable. The large deviation formulae valid outside the bulk are also derived. Obtained formulae agree well with the results of direct numerical calculations.